A014062 -id:A014062 - cp's OEIS frontend

A137610 Self-convolution of A014062, where A014062(n) = C(n^2, n).

1, 2, 13, 180, 3844, 110908, 4030740, 176640072, 9059743648, 532179428700, 35219852623888, 2592514449263656, 210072673380786552, 18579938909696752728, 1780987027765227959096, 183907984490301947455872

Offset: 0

Paul D. Hanna, Jan 29 2008

Cf. A014062.

Table[Sum[Binomial[k^2, k]*Binomial[(n-k)^2, n-k], {k,0,n}], {n,0,20}] (* Vaclav Kotesovec, Aug 20 2025 *)

a(n)=sum(k=0,n,binomial(k^2,k)*binomial((n-k)^2,n-k))

a(n) = Sum_{k=0..n} C(k^2, k) * C((n-k)^2, n-k).

a(n) ~ sqrt(2) * exp(n - 1/2) * n^(n - 1/2) / sqrt(Pi). - Vaclav Kotesovec, Aug 20 2025

A167010 a(n) = Sum_{k=0..n} C(n,k)^n.

Original entry on oeis.org

1, 2, 6, 56, 1810, 206252, 86874564, 132282417920, 770670360699138, 16425660314368351892, 1367610300690018553312276, 419460465362069257397304825200, 509571049488109525160616367158261124, 2290638298071684282149128235413262383804352

Offset: 0

Paul D. Hanna, Nov 17 2009

The number of n*n 0-1 matrices with equal numbers of nonzeros in every row. - David Eppstein, Jan 19 2012

			The triangle A209427 of coefficients C(n,k)^n, n>=k>=0, begins:
  1;
  1,     1;
  1,     4,        1;
  1,    27,       27,        1;
  1,   256,     1296,      256,        1;
  1,  3125,   100000,   100000,     3125,     1;
  1, 46656, 11390625, 64000000, 11390625, 46656,    1; ...
in which the row sums form this sequence.

Vincenzo Librandi, Table of n, a(n) for n = 0..59
Vaclav Kotesovec, Interesting asymptotic formulas for binomial sums, Jun 09 2013.

Cf. A000312, A014062, A066300, A167009, A167007, A209427, A218792.

[(&+[Binomial(n,j)^n: j in [0..n]]): n in [0..20]]; // G. C. Greubel, Aug 26 2022

Table[Sum[Binomial[n, k]^n, {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Nov 05 2012 *)

PARI
```
a(n)=sum(k=0,n,binomial(n,k)^n)
```

[sum(binomial(n,j)^n for j in (0..n)) for n in (0..20)] # G. C. Greubel, Aug 26 2022

Ignoring initial term, equals the logarithmic derivative of A167007. [Paul D. Hanna, Nov 18 2009]
If n is even then a(n) ~ c * exp(-1/4) * 2^(n^2 + n/2)/((Pi*n)^(n/2)), where c = Sum_{k = -oo..oo} exp(-2*k^2) = 1.271341522189... (see A218792). - Vaclav Kotesovec, Nov 05 2012
If n is odd then c = Sum_{k = -infinity..infinity} exp(-2*(k+1/2)^2) = 1.23528676585389... - Vaclav Kotesovec, Nov 06 2012
a(n) = (n!)^n * [x^n] (Sum_{k>=0} x^k / (k!)^n)^2. - Ilya Gutkovskiy, Jul 15 2020

A060539 Table by antidiagonals of number of ways of choosing k items from n*k.

Original entry on oeis.org

1, 1, 2, 1, 6, 3, 1, 20, 15, 4, 1, 70, 84, 28, 5, 1, 252, 495, 220, 45, 6, 1, 924, 3003, 1820, 455, 66, 7, 1, 3432, 18564, 15504, 4845, 816, 91, 8, 1, 12870, 116280, 134596, 53130, 10626, 1330, 120, 9, 1, 48620, 735471, 1184040, 593775, 142506, 20475, 2024, 153, 10

Offset: 1

Henry Bottomley, Apr 02 2001

			Square array A(n,k) begins:
  1,  1,    1,     1,      1,       1,        1, ...
  2,  6,   20,    70,    252,     924,     3432, ...
  3, 15,   84,   495,   3003,   18564,   116280, ...
  4, 28,  220,  1820,  15504,  134596,  1184040, ...
  5, 45,  455,  4845,  53130,  593775,  6724520, ...
  6, 66,  816, 10626, 142506, 1947792, 26978328, ...
  7, 91, 1330, 20475, 324632, 5245786, 85900584, ...

Seiichi Manyama, Antidiagonals n = 1..140, flattened (first 20 antidiagonals from Harry J. Smith)
H. J. Brothers, Pascal's Prism: Supplementary Material.

Rows include A000012, A000984, A005809, A005810, A001449, A004355, A004368.
Columns include A000027, A000384, A006566, A060541.
Main diagonal is A014062.
Cf. A295772.

A:= (n, k)-> binomial(n*k, k):
seq(seq(A(n, 1+d-n), n=1..d), d=1..10);  # Alois P. Heinz, Jul 28 2023

{ i=0; for (m=1, 20, for (n=1, m, k=m - n + 1; write("b060539.txt", i++, " ", binomial(n*k, k))); ) } \\ Harry J. Smith, Jul 06 2009

A(n,k) = binomial(n*k,k) = A007318(n*k,k) = A060538(n,k)/A060538(n-1,k).

A167008 a(n) = Sum_{k=0..n} C(n,k)^k.

Original entry on oeis.org

1, 2, 4, 14, 106, 1732, 66634, 5745700, 1058905642, 461715853196, 461918527950694, 989913403174541980, 5009399946447021173140, 60070720443204091719085184, 1548154498059133199618813305334, 92346622775540905956057053976278584

Offset: 0

Paul D. Hanna, Nov 17 2009

Row sums of A219206.

Reinhard Zumkeller, Table of n, a(n) for n = 0..75
Vaclav Kotesovec, Interesting asymptotic formulas for binomial sums, Jun 09 2013.

Cf. A000169, A014062, A167009, A167010, A184731, A219206, A220359, A362288.

a167008 = sum . a219206_row  -- Reinhard Zumkeller, Feb 27 2015

[(&+[Binomial(n,j)^j: j in [0..n]]): n in [0..20]]; // G. C. Greubel, Aug 26 2022

Flatten[{1,Table[Sum[Binomial[n, k]^k, {k,0,n}], {n,20}]}]
(* Program for numerical value of the limit a(n)^(1/n^2) *) (1-r)^(-r/2)/.FindRoot[(1-r)^(2*r-1)==r^(2*r),{r,1/2},WorkingPrecision->100] (* Vaclav Kotesovec, Dec 12 2012 *)
Total/@Table[Binomial[n,k]^k,{n,0,20},{k,0,n}] (* Harvey P. Dale, Oct 19 2021 *)

PARI
```
a(n)=sum(k=0,n,binomial(n,k)^k)
```

[sum(binomial(n,j)^j for j in (0..n)) for n in (0..20)] # G. C. Greubel, Aug 26 2022

Limit_{n->oo} a(n)^(1/n^2) = (1-r)^(-r/2) = 1.533628065110458582053143..., where r = A220359 = 0.70350607643066243... is the root of the equation (1-r)^(2*r-1) = r^(2*r). - Vaclav Kotesovec, Dec 12 2012

A167009 a(n) = Sum_{k=0..n} C(n^2, n*k).

Original entry on oeis.org

1, 2, 8, 170, 16512, 6643782, 11582386286, 79450506979090, 2334899414608412672, 265166261617029717011822, 128442558588779813655233443038, 238431997806538515396060130910954852

Offset: 0

Paul D. Hanna, Nov 17 2009

nonn

			The triangle A209330 of coefficients C(n^2, n*k), n>=k>=0, begins:
  1;
  1,       1;
  1,       6,          1;
  1,      84,         84,          1;
  1,    1820,      12870,       1820,          1;
  1,   53130,    3268760,    3268760,      53130,       1;
  1, 1947792, 1251677700, 9075135300, 1251677700, 1947792,     1; ...
in which the row sums form this sequence.

Vincenzo Librandi, Table of n, a(n) for n = 0..58
Vaclav Kotesovec, Interesting asymptotic formulas for binomial sums, Jun 09 2013.

Cf. A014062, A167006, A167010, A209330, A218792, A306846.

[(&+[Binomial(n^2, n*j): j in [0..n]]): n in [0..20]]; // G. C. Greubel, Aug 26 2022

Table[Sum[Binomial[n^2,n*k],{k,0,n}],{n,0,15}] (* Harvey P. Dale, Dec 11 2011 *)

PARI
```
a(n)=sum(k=0,n,binomial(n^2,n*k))
```

[sum(binomial(n^2, n*j) for j in (0..n)) for n in (0..20)] # G. C. Greubel, Aug 26 2022

Ignoring initial term, equals the logarithmic derivative of A167006. - Paul D. Hanna, Nov 18 2009
If n is even then a(n) ~ c * 2^(n^2 + 1/2)/(n*sqrt(Pi)), where c = Sum_{k = -infinity..infinity} exp(-2*k^2) = 1.271341522189... (see A218792). - Vaclav Kotesovec, Nov 05 2012
If n is odd then c = Sum_{k = -infinity..infinity} exp(-2*(k+1/2)^2) = 1.23528676585389... - Vaclav Kotesovec, Nov 06 2012
a(n) = A306846(n^2,n) = [x^(n^2)] (1-x)^(n-1)/((1-x)^n - x^n) for n > 0. - Seiichi Manyama, Oct 11 2021

A090642 Triangle read by rows: T(n,k) = binomial(n^2, k), 0 <= k <= n.

Original entry on oeis.org

1, 1, 1, 1, 4, 6, 1, 9, 36, 84, 1, 16, 120, 560, 1820, 1, 25, 300, 2300, 12650, 53130, 1, 36, 630, 7140, 58905, 376992, 1947792, 1, 49, 1176, 18424, 211876, 1906884, 13983816, 85900584, 1, 64, 2016, 41664, 635376, 7624512, 74974368, 621216192, 4426165368

Offset: 0

Reinhard Zumkeller, Dec 13 2003

A066382(n) = Sum_{k=0..n} T(n,k).

			Triangle begins:
  1;
  1,  1;
  1,  4,   6;
  1,  9,  36,   84;
  1, 16, 120,  560,  1820;
  1, 25, 300, 2300, 12650,  53130;
  1, 36, 630, 7140, 58905, 376992, 1947792;
  ...

Nathaniel Johnston, Rows 0..100, flattened

Cf. A007318 (Pascal's triangle), A014062 (right diagonal).

for n from 0 to 6 do seq(binomial(n^2,k),k=0..n); od; # Nathaniel Johnston, Jun 24 2011

A096131 Sum of the terms of the n-th row of triangle pertaining to A096130.

Original entry on oeis.org

1, 7, 105, 2386, 71890, 2695652, 120907185, 6312179764, 375971507406, 25160695768715, 1869031937691061, 152603843369288819, 13584174777196666630, 1309317592648179024666, 135850890740575408906465

Offset: 1

Amarnath Murthy, Jul 04 2004

nonn

The product of the terms of the n-th row is given by A034841.

Collection of partial binary matrices: 1 to n rows of length n and a total of n entries set to one in each partial matrix. - Olivier Gérard, Aug 08 2016

			From _Seiichi Manyama_, Aug 18 2018: (Start)
a(1) = (1/1!) * (1) = 1.
a(2) = (1/2!) * (1*2 + 3*4) = 7.
a(3) = (1/3!) * (1*2*3 + 4*5*6 + 7*8*9) = 105.
a(4) = (1/4!) * (1*2*3*4 + 5*6*7*8 + 9*10*11*12 + 13*14*15*16) = 2386. (End)

Seiichi Manyama, Table of n, a(n) for n = 1..338

Cf. A014062, A096130, A034841, A007318, A226391, A167009, A167008, A167010, A072034, A086331, A349470.

List(List([1..20],n->List([1..n],k->Binomial(k*n,n))),Sum); # Muniru A Asiru, Aug 12 2018

A096130 := proc(n,k) binomial(k*n,n) ; end: A096131 := proc(n) local k; add( A096130(n,k),k=1..n) ; end: for n from 1 to 18 do printf("%d, ",A096131(n)) ; od ; # R. J. Mathar, Apr 30 2007
seq(add((binomial(n*k,n)), k=0..n), n=1..15); # Zerinvary Lajos, Sep 16 2007

Table[Sum[Binomial[k*n, n], {k, 0, n}], {n, 1, 20}] (* Vaclav Kotesovec, Jun 06 2013 *)

a(n) = sum(k=1, n, binomial(k*n, n)); \\ Michel Marcus, Aug 20 2018

a(n) = Sum_{k=1..n} binomial(k*n, n). - Reinhard Zumkeller, Jan 09 2005
a(n) = (1/n!) * Sum_{j=1..n} Product_{i=n*(j-1)+1..n*j} i. - Reinhard Zumkeller, Jan 09 2005 [corrected by Seiichi Manyama, Aug 17 2018]
a(n) ~ exp(1)/(exp(1)-1) * binomial(n^2,n). - Vaclav Kotesovec, Jun 06 2013

More terms from R. J. Mathar, Apr 30 2007

Edited by N. J. A. Sloane, Sep 06 2008 at the suggestion of R. J. Mathar

A135860 a(n) = binomial(n*(n+1), n).

Original entry on oeis.org

1, 2, 15, 220, 4845, 142506, 5245786, 231917400, 11969016345, 706252528630, 46897636623981, 3461014728350400, 281014969393251275, 24894763097057357700, 2389461906843449885700, 247012484980695576597296, 27361230617617949782033713, 3233032526324680287912449550

Offset: 0

Paul D. Hanna, Dec 02 2007

Seiichi Manyama, Table of n, a(n) for n = 0..337
R. Mestrovic, Wolstenholme's theorem: Its Generalizations and Extensions in the last hundred and fifty years (1862-2011), arXiv:1111.3057 [math.NT], 2011.

Cf. A014062, A014068, A054688, A107863, A135861, A135862.

[Binomial(n*(n+1), n): n in [0..30]]; // G. C. Greubel, Feb 20 2022

Table[Binomial[n^2 + n, n], {n, 0, 16}] (* Arkadiusz Wesolowski, Jul 18 2012 *)
(* or *)
Table[SeriesCoefficient[(1+x)^(n*(n+1)), {x, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Aug 06 2025 *)

a(n)=binomial(n*(n+1),n)
for(n=0,15,print1(a(n),", "))

a(n)=sum(k=0,n,binomial(n,k)*binomial(n^2,k))
for(n=0,15,print1(a(n),", "))

[binomial(n*(n+1), n) for n in (0..30)] # G. C. Greubel, Feb 20 2022

a(n) = Sum_{k=0..n} binomial(n,k) * binomial(n^2,k). - Paul D. Hanna, Nov 18 2015
a(n) is divisible by (n+1): a(n)/(n+1) = A135861(n).
a(n) is divisible by (n^2+1): a(n)/(n^2+1) = A135862(n).
a(n) = binomial(2*A000217(n),n). - Arkadiusz Wesolowski, Jul 18 2012
a(n) = [x^n] 1/(1 - x)^(n^2+1). - Ilya Gutkovskiy, Oct 03 2017
a(n) ~ exp(n + 1/2) * n^(n - 1/2) / sqrt(2*Pi). - Vaclav Kotesovec, Feb 08 2019
a(p) == p + 1 ( mod p^4 ) for prime p >= 5 and a(2*p) == (4*p + 1)*(2*p + 1) ( mod p^4 ) for all prime p. Apply Mestrovic, equation 37. - Peter Bala, Feb 27 2020
a(n) = ((n^2 + n)!)/((n^2)! * n!). - Peter Luschny, Feb 27 2020
a(n) = [x^n] (1 + x)^(n*(n+1)). - Vaclav Kotesovec, Aug 06 2025

A177234 a(n) = binomial(n^2, n)/(n+1).

Original entry on oeis.org

2, 21, 364, 8855, 278256, 10737573, 491796152, 26088783435, 1573664496040, 106395830418878, 7970714909592876, 655454164338881388, 58702034425556612832, 5687847988198592380965, 592867741295430227919600

Offset: 2

Michel Lagneau, May 05 2010, May 08 2010

nonn

Theorem: binomial(n^2, n)/(n+1) is an integer for n >= 2.
Proof 1 from William J. Keith, May 08 2010:
binomial(n^2, n) * 1/(n+1)
= (n^2)*(n^2-1)*(n^2-2)!/((n^2-n)!*n*(n-1)*(n-2)!) * 1/(n+1)
= n*(n^2-2)!/((n^2-n)!*(n-2)!) = n * binomial(n^2-2,n-2). QED
Proof 2 from Max Alekseyev, May 08 2010:
Recall that the valuation of m! w.r.t. prime p equals the sum floor(m/p^i) over i=1,2,3,...
Moreover, if m=a+b where a and b are nonnegative integers, then floor(m/p^i) - floor(a/p^i) - floor(b/p^i) >= 0.
Let n>1. To prove that binomial(n^2, n)/(n+1) is an integer, it is enough to show that its valuation w.r.t. any prime p is nonnegative.
It is clear that trouble may come only from primes dividing n+1.
Let valuation(n+1,p)=k > 0, i.e., n+1=p^k*m where prime p does not divide m.
Then n = p^k*m - 1, n^2 = p^(2k)*m^2 - 2*p^k*m + 1 and n^2 - n = p^(2k)*m^2 - 3*p^k*m + 2.
It is easy to check that floor(n^2/p^i) - floor(n/p^i) - floor((n^2-n)/p^i) = 1 for i=1,2,...,k if p>2 and for i=2,3,...,k+1 if p=2, implying that valuation(binomial(n^2, n)/(n+1),p) >= 0. QED

			a(3) = 21 because binomial(9,3)/(3+1) = 84/4 = 21.

H. Gupta and S. P. Khare, On C(k^2,k) and the product of the first k primes, Publ. Fac. Electrotechn. Belgrade, Ser. Math. Phys. 25-29 (1977) 577-598.

G. C. Greubel, Table of n, a(n) for n = 2..335
H. Gupta and S. P. Khare, On C(k^2,k) and the product of the first k primes, Publ. Fac. Electrotechn. Belgrade, Ser. Math. Phys. 25-29 (1977) 577-598. [PDF] [From _R. J. Mathar_, May 09 2010]

Cf. A014062 A123312, A177456, A177784, A177788.

[Binomial(n^2,n)/(n+1): n in [2..30]]; // G. C. Greubel, Apr 27 2024

with(numtheory):n0:=25:T:=array(1..n0-1):for n from 2 to n0 do: T[n-1]:= binomial(n*n,n)/(n+1):od:print(T):

Table[Binomial[n^2,n]/(n+1), {n,2,30}] (* G. C. Greubel, Apr 27 2024 *)

[binomial(n^2,n)/(n+1) for n in range(2,31)] # G. C. Greubel, Apr 27 2024

From G. C. Greubel, Apr 27 2024: (Start)
a(n) = A014062(n)/(n+1).
a(n) = A177456(n)/(n-1).
a(n) = n*A177784(n).
a(n) = A177788(n)/n. (End)

A209330 Triangle defined by T(n,k) = binomial(n^2, n*k), for n>=0, k=0..n, as read by rows.

Original entry on oeis.org

1, 1, 1, 1, 6, 1, 1, 84, 84, 1, 1, 1820, 12870, 1820, 1, 1, 53130, 3268760, 3268760, 53130, 1, 1, 1947792, 1251677700, 9075135300, 1251677700, 1947792, 1, 1, 85900584, 675248872536, 39049918716424, 39049918716424, 675248872536, 85900584, 1, 1

Offset: 0

Paul D. Hanna, Mar 06 2012

Column 1 equals A014062.
Row sums equal A167009.
Antidiagonal sums equal A209331.
Ignoring initial row T(0,0), equals the logarithmic derivative of the g.f. of triangle A209196.

			The triangle of coefficients C(n^2,n*k), n>=k, k=0..n, begins:
1;
1, 1;
1, 6, 1;
1, 84, 84, 1;
1, 1820, 12870, 1820, 1;
1, 53130, 3268760, 3268760, 53130, 1;
1, 1947792, 1251677700, 9075135300, 1251677700, 1947792, 1;
1, 85900584, 675248872536, 39049918716424, 39049918716424, 675248872536, 85900584, 1; ...

Paul D. Hanna, Rows n = 0..30, as a flattened table of n, a(n) for n = 0..495

Cf. A014062 (column 1), A167009 (row sums), A209331, A209196.
Cf. related triangles: A209196 (exp), A228836, A228832, A226234.
Cf. A206830.

Table[Binomial[n^2, n*k], {n,0,10}, {k,0,n}]//Flatten (* G. C. Greubel, Jan 05 2018 *)

{T(n,k)=binomial(n^2,n*k)}
for(n=0,10,for(k=0,n,print1(T(n,k),", "));print(""))

cp's OEIS Frontend

A137610 Self-convolution of A014062, where A014062(n) = C(n^2, n).

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A167010 a(n) = Sum_{k=0..n} C(n,k)^n.

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A060539 Table by antidiagonals of number of ways of choosing k items from n*k.

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A167008 a(n) = Sum_{k=0..n} C(n,k)^k.

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A167009 a(n) = Sum_{k=0..n} C(n^2, n*k).

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A090642 Triangle read by rows: T(n,k) = binomial(n^2, k), 0 <= k <= n.

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Maple

A096131 Sum of the terms of the n-th row of triangle pertaining to A096130.

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